Prospective Elementary and Secondary Teachers' Understanding of DivisionAuthor(s): Deborah Loewenberg BallSource: Journal for Research in Mathematics Education, Vol. 21, No. 2 (Mar., 1990), pp. 132-144Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/749140Accessed: 25/10/2009 13:01Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available athttp:www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unlessyou have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and youmay use content in the JSTOR archive only for your personal, non-commercial use.Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained athttp:www.jstor.org/action/showPublisher?publisherCode=nctm.Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printedpage of such transmission.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to Journal for Research in Mathematics Education.http://www.jstor.org
Journal for Research in Mathematics Education 1990, Vol. 21, No. 2, 132-144 PROSPECTIVE ELEMENTARY AND SECONDARY TEACHERS' UNDERSTANDING OF DIVISION DEBORAH LOEWENBERG BALL, Michigan State University This article reports an analysis of 19 prospective elementary and secondary teachers' understanding of division. Interview questions probed the prospective teachers' understanding of division in three contexts. Although many of the teacher candidates could produce correct answers, several could not, and few were able to give mathematical explanations for the underlying principles and meanings. The prospective teachers' knowledge was generally fragmented, and each case of division was held as a separate bit of knowledge. Teachers' subject matter knowledge is enjoying a renaissance in research on teaching (Shulman, 1986) and teacher education (National Center for Research on Teacher Education, 1988). From a range of perspectives and with a variety of approaches, researchers are focusing increasingly on the subject matter knowledge of teachers and its role in teaching. In research on the teaching of mathematics, some researchers investigate teachers' and prospective teachers' beliefs about the subject or their ideas about teaching it (Blaire, 1981; Ernest, 1988; Ferrini-Mundy, 1986; Kuhs, 1980; Lerman, 1983; Peterson, Fennema, Carpenter, & Loef, 1989; Thompson, 1984). These studies generally highlight the influence of teachers' assumptions about mathematics on their teaching of the subject. Other researchers focus on teachers' and prospective teachers' understandings of specific topics (Ball, 1988; Ball & McDiarmid, 1988; Davis, 1986; Even, 1989; Leinhardt & Smith, 1985; Owens, 1987; Post, Behr, Harel, & Lesh, 1988; Steinberg, Haymore, & Marks, 1985). They explore how teachers think about their mathematical knowl- edge and how they understand (or misunderstand) specific ideas. The analysis reported in this article is part of a larger study that examined pro- spective teachers' knowledge and reasoning, both mathematical and pedagogical, at the point when they entered teacher education (Ball, 1988). This article focuses on one aspect of the teacher candidates' substantive knowledge of mathematics, their understanding of division. The article also examines what they believed makes something true or reasonable in mathematics, what counts as a mathemati- cal justification. This research was supported in part by the National Center for Research on Teacher Edu- cation, Michigan State University. The NCRTE is funded primarily by the Office of Educa- tional Research and Improvement, United States Department of Education. The opinions expressed herein are those of the author and do not necessarily reflect the position, policy, or endorsement of the OERI/ED. (Grant No. OERI-G-86-0001) The author gratefully acknowledges Ruhama Even, Mary M. Kennedy, Magdalene Lam- pert, Penelope Peterson, and Suzanne M. Wilson, as well as the JRME editor and reviewers, for their helpful comments on this article.
133 METHOD Participants Ten elementary and nine secondary education students were interviewed when they were about to enroll for their first education course. The prospective elemen- tary teachers were majoring in elementary education or child development and had no disciplinary specialization. The secondary teacher education students were mathematics majors or minors. The sample of prospective teachers was systematically selected to vary on sev- eral key criteria: gender, academic history in college mathematics, and self-ac- knowledged dispositions toward mathematics. Of the 19 students, 6 were men and 13 were women. One student was black, one was Asian; the others were Cauca- sian. Although the group was not selected to be representative, the demographic data indicated that the sample was similar to the population of teacher education students in terms of characteristics such as ethnicity, gender, social class, and age. Student academic records were an additional source of data about the prospec- tive teachers selected to participate in the study. Analyses of academic records revealed three high school valedictorians within the sample (one elementary and two secondary majors). On a 4-point scale, the average overall grade-point aver- age of the elementary majors was 3.18; the secondary majors averaged 3.09. The mean grade-point average for mathematics courses was 3.12 and 3.13, respec- tively. The elementary teacher candidates had taken, on average, fewer than two mathematics courses since high school, and these were often remedial. The secon- dary teacher candidates had taken an average of 32 quarter credits, or more than nine college-level mathematics courses. Instrumentation Three different mathematical contexts were employed to examine prospective teachers' knowledge of division: division with fractions, division by zero, and division with algebraic equations. In each case, the teacher candidates were asked to explain or to generate representations. Probes were developed to examine the teacher candidates' ideas about what it means to justify or to explain something in mathematics. The three questions were posed in the order reported in this article and were not separated by other questions or tasks. The interview questions and standard probes are reported below. Additional probes were generated in the course of the individual interviews when it was nec- essary in order to get clearer about what the person was thinking. In all cases, ef- fort was directed at probing as deeply as possible the prospective teacher's under- standing of the mathematical ideas and of what counts as a mathematically reason- able justification. Procedures The interviews were tape-recorded and transcribed. Careful substantive analy- ses of the interview questions led to the creation of a set of response categories for
134 Prospective Teachers' Understanding of Division each one. These categories were modified in the course of data analysis to better accommodate teacher candidates' responses. In the larger study, most questions were cross-analyzed on several dimensions: subject matter understanding; ideas about teaching; learning and the teacher's role; and feelings or attitudes about mathematics, pupils, or self. For the present analysis of prospective teachers' knowledge of division, responses were coded on two dimensions: (1) their correct- ness and (2) the nature of the justification provided by the teacher candidate. RESULTS Division By Fractions Participants were asked the following: People have different approaches to solving problems involving division with frac- tions. How would you solve this one: Sometimes teachers try to come up with real-world situations or story problems to show the meaning or application of some particular piece of content. This can be pretty challenging to do. What would you say would be a good situation or story for 1/4 + 2-something real for which 1/4 + /2 is the appropriate mathematical formulation? After the participant described a situation or story, he or she was asked: How does that fit with 13A + /A? If the participant noticed that the answer to the story or other representation did not match the answer obtained computationally, he or she was asked: Why did that come out different? If the participant struggled with finding a representation, or could not do it, he or she was asked: Many people find this hard. In your view, what makes this difficult? The teacher candidates' responses to being asked to solve and to develop a rep- resentation for 13/4 - 1/2 suggested that they perceived the task to be about fractions, not division. When asked, for example, what made this difficult, most commented that it was hard (or impossible) to relate 1/4 + 1/2 to real life because, as one said, "you don't think in fractions, you think more in whole numbers." Not only did their explanations reveal that they framed the problem in terms of fractions, but also that many were uncomfortable with fractions as quantities. Several com- mented that they did not like fractions. Seventeen out of 19 participants were able to calculate the division correctly, but only 5 were able to generate an appropriate representation. Their responses are summarized in Table 1. Appropriate representations. Five teacher candidates, all mathematics majors, generated an appropriate representation of 1/4 + V2; however, this did not come easily to any of them. For example, Terrell first said he couldn't "think of anything
Deborah Loewenberg Ball 135 Table 1 Prospective Elementary and Secondary Teachers' Calculations and Representations of 1 3/4 + 1/2 Elementary Secondary Totals Correct calculation 9 8 17 Appropriate representation 0 5 5 Inappropriate representation 3 2 5 Unable to generate a representation 6 2 8 Note. There were 10 elementary and 9 secondary teacher candidates. One elementary candidate was not asked this question. specific." Then he said he would use pizza: If you took the pizza and took one-half of a pizza and you took a whole pizza and three- quarters of a pizza, that would be 11/4. You put the one-half of the pizza on top of each piece. So first you'd take the whole pizza and you'd put it on top of it. Then you'd take that off, whatever it fits on and you'd do it again. Only take it off if it fits the whole thing. If...both pieces are equal. Then you go through the one-half a piece and do the same for that. Take that off. Then you get that last piece and you...well, that's the way I'd explain it. Terrell then explained what the answer (31/2) meant in this context: You'd take the one-half and the answer would be how many times you got a whole half (if you want to say that). Of the...whatever's left over, what part of it is of the half, I guess you could say. You'd have a V4 left, which is one-half of a half. While sometimes rather confusing to follow, these prospective teachers' re- sponses did make mathematical sense. Their answers indicated that, using the measurement model of division, they were thinking in terms of how many halves there are in one and three-fourths. Inappropriate representations. Five teacher candidates generated representa- tions that did not correspond to 1 /4+ ?. The most frequent error was to represent division by two instead of division by one-half. For example, Barb, a mathematics major, gave the following story: If we had one and three-quarters pizzas left and there were two of us dying to split it, then how would we be able to split that? Answering her own questions, she said each person would get 7/8 of a pizza alto- gether. This error, representing division by two rather than division by one-half, was the most common error made by the participants. Although the answer they got to their story problem (7/8) differed from their cal- culated answer (3?), the four prospective teachers did not seem to notice that they were dividing in half instead of by one-half. The discrepancy seemed to be masked for them by a change in the referent unit from whole pizzas to fourths of pizzas. Here is a typical example of the students' reasoning: "I have 13/ pizzas which I want to split equally between two hungry teenagers (1/4 + 2). Each pizza is divided into four pieces, so you have seven pieces. So each person gets 7/ of a pizza, which is 3? pieces of pizza." Allen, an elementary major with 27 credits in college mathematics (through the
136 Prospective Teachers' Understanding of Division calculus course sequence) seemed to get stuck by his knowlege of the algorithm "invert and multiply": Somebody has one and three quarters apples or something like that and they wanted to double up their, double the amount of apples they have, um, just give 'em an equation...using only fractions. Other than that I couldn't think of any, any situation where you could, where you would logically divide a number like that by one half instead of just multiplying by two. You would have to have, be, be working on, on dividing like fractions and setting up equations using that. Right, you just have to say...."Well, you know, use this and figure the story problem out but, only use frac- tions." Allen's story modeled 1/4 x 2-the common procedure used in fraction division. Using "invert and multiply" as his frame of reference, Allen did not seem to focus on the concept of division by one-half. No representation. Eight teacher candidates could not generate any representa- tion, correct or incorrect, for 13/4 + 1/2. Marsha, for example, said she felt stuck and couldn't even remember how to get the answer. She explained that she hadn't done this since high school: Egad, I don't even remember how to do this. (pause) Shoot, no, I have to, I don't know how to do it. Because I don't know what I did ...I found the common denominator and I think what I have to do is go four times one is four and then plus three is seven- fourths, no I think that's what I did, one-half, but then, see, I don't know what I need to divide. I don't even remember that...I remember doing these for a long time though and trying to get these down, and so I remember bits and pieces and then I try to apply it generally, and I can't do it. The prospective teachers who did not generate a representation at all seemed to fall into two groups. Two participants (one elementary, one secondary) did recog- nize the conceptual problem. They initially proposed stories or models that repre- sented division by two and then realized that they were representing division in half, not by one-half. The other six participants, however, seemed to think that trying to relate 1/4 + V2 to a concrete situation was not a feasible task-that 1/4 + /2 could not be represented in real-world terms. The two participants who recognized that division by one-half is not the same as division by two revealed a better grasp of the idea than those who constructed a story that represented division by two. Still, despite this recognition, they were unable to figure out what division by one-half meant. Those who thought it was an impossible task may have been revealing a view of mathematics as a senseless activity, out of which meaning cannot necessarily be made. Division by Zero Participants were asked the following questions: Suppose that a student asks you what 7 divided by 0 is. How would you respond? Why is that what you'd want to say? Following whatever the participant said, he or she was probed: Suppose the student asks you why that is? If a participant said, "I'd say it's undefined," he or she was asked:
Deborah Loewenberg Ball 137 What if the student asks, "What do you mean by 'undefined'?" If a participant said, "I'd say you can't divide by 0," he or she was asked: What if a student asks, "Why can't you divide by zero?" If a participant said they would show students how, as you divide by smaller and smaller numbers, the answer gets larger and larger, he or she was asked: What would I see or hear you doing? Of the 19 teacher candidates, 5 explained the meaning of division by zero in response to this question. Twelve of the prospective teachers responded by stating rules, five of which were incorrect. Two participants did not know. (See Table 2 for the distribution of types of responses.) Each category is discussed below. Table 2 Prospective Elementary and Secondary Teachers' Explanations for Division by Zero Elementary Secondary Totals Meaning 1 4 5 Correct rule 2 5 7 Incorrect rule 5 0 5 Don't know 2 0 2 Note. There were 10 elementary and 9 secondary teacher candidates. Explanations focused on meaning. Four teacher candidates gave answers that fo- cused on what division by zero means. Two approaches were used: (1) showing that division by zero was undefined and (2) showing that the quotient "explodes" as the divisor decreases. Tim, a mathematics major, chose the first approach. He said that he would write 7 + 0 "in mathematical form" on the board-i.e., with a division bracket: 0)7. Then he would explain that: You cannot divide seven by zero because there is nothing multiplied by zero to get seven. In other words, everything multiplied by a zero is zero, so if we had seven over zero. Okay, if we had zero divided by seven there is nothing multiplied, there is no number up here you could put to get zero. There's no number you can put up here to get seven. And I would show them that. Whereas six divided by two there's a number you can put up there. And whenever you come across that case, you can't find a num- ber to put up there, it doesn't exist, you can't do it. Allen, an elementary major, explained division by zero using the second ap- proach: Dividing seven by three and then divide seven by two and then divide seven by one and, uh, and when they get up to seven divided by one is seven and you were to go one step farther you'd, you'd have numbers that were keep getting larger and larger....Um, one step farther you would have to, you know, say divide it by zero....I guess it would be better to start getting closer to zero using the decimals and see that dividing seven by fractions makes numbers, you know, they keep getting larger and larger and, uh, that if you keep making that, the divisor closer and closer to zero, the number's just gonna keep getting larger and larger and larger and, uh, then I'd start asking them what the largest number they can think of is so then, that there is no largest number that, uh, that it, there is really no such statement as seven divided by zero. Mark and Allen both focused specifically on the case of dividing by zero. What
138 Prospective Teachers' Understanding of Division their answers had in common was the aim of showing why the particular case of division by zero is impossible. You can't divide by zeto. Seven teacher candidates explained division by zero in terms of a rule such as "you can't divide by zero." Unlike those who focused on meaning, these prospective teachers did not try to show why this was so. Instead they emphasized the importance of remembering the rule. Terrell, a mathematics major, said emphatically, I'd just say..."It's undefined," and I'd tell them that this is a rule that you should never forget that anytime you divide by zero, you can't. You just can't. It's undefined, so...you just can't. He added, "Anytime you get a number divided by zero, then you did something wrong before." Andy, another mathematics major, said, "You can't divide by zero...It's just something to remember." Cindy, also majoring in mathematics, said she would tell students that "this is something that you won't ever be able to do in mathematics"--even in calculus. Even when probed, these students did not provide a mathematical justification for the principle that division by zero was not permitted. Anything divided by zero is zero. Five other teacher candidates also responded in term of a rule. Like the prospective teachers quoted above, their notions of mathematical explanation seemed to mean restating rules. What made their re- sponses different, however, was that the rule they invoked was not true. Linda, an elementary major, was perhaps the most emphatic: I'd just say, "Anything divided by zero is zero. That's just a rule, you just know it." Or I'd say, "Well, if you don't have anything, you can't get anything out. You know, it's empty, it's nothing, so you can't get anything out of it." Anything multiplied by zero is zero. I'd just say, "That's something that you have to learn, you have to know." I think that's how I was told. You just know it...I'd just say, you know, if they were older and they asked me "Why?" I'd just have to start mumbling about something, I don't know...I don't know what. I'd just tell them "Because!" (laughs) That's just the way it is...that's just one of those rules...something like that...you know, in English...sometimes the C sounds like K or...you know, you just learn. I before E ex- cept after C, one of those things, in my view. Interestingly, although Linda mentioned multiplication by zero, she didn't con- nect it with the problem of dividing by zero. Like those who stated "you can't divide by zero," these prospective teachers all emphasized the absoluteness of the rule and the value of getting pupils to remem- ber it. Explaining and knowing mathematics was centered on rules. However, these teacher candidates did not realize that their rules were invalid. I don't remember. Two prospective elementary teachers said they could not remember the answer to 7 + 0. Mei Ling said simply, "Seven divided by zero? Isn't that-isn't there a term for the answer to that? I can't remember." Rachel, who had taken a little more mathematics, more recently and more successfully than most of the other elementary majors, was simply stumped by this question. "Seven divided by zero," she mused. "I'm having trouble...is that zero or is that seven? I'm trying to think myself." Because these students could not remember the answer
Deborah Loewenberg Ball 139 to 7 + 0, their ideas about what would count as a reasonable explanation were not revealed. Algebraic Equations Participants were asked: Suppose that one of your students asks you for help with the following: x If = 5, then x = 0.2 How would you respond? Why is that what you'd do? If a participant said, "I'd tell them to do the same thing to both sides," or "I'd tell them to multiply both sides by 0.2," he or she was asked: What if the student asks, "Why do you do that?" In response to this question involving division in algebra, the teacher candidates focused on the mechanics of solving algebraic equations. Overwhelmingly they "explained" x -=5 0.2 by restating the steps of procedures to solve such equations. Only one prospective teacher talked about it in terms of what it meant, and a few teacher candidates didn't know how to do it at all. (See Table 3.) Table 3 Prospective Elementary and Secondary Teachers' Explanations for Division in Algebra Elementary Secondary Totals Meaning 1 0 1 Procedure 5 9 14 Don't know 4 0 4 Note. There were 10 elementary and 9 secondary teacher candidates. Focus on meaning. Only one teacher candidate-an elementary major-tried to talk about the meaning of the equation. Sandi said that she would want the pupil "to understand what he's doing first." She said she would help the pupil under- stand "the idea that the .2 has to go into x." Although her explanation was vague, she was trying to make sense of the problem by reasoning about division. Focus on mechanics. Fourteen of the prospective teachers, including all of the mathematics majors, focused on the mechanics of manipulating algebraic equa- tions. Terrell, a secondary candidate, said I'd explain that somehow you have to get this x by itself without that .02, I mean 0.2...and then I'd ask her, I'd ask her...I'd tell her somehow she's going to have to get rid of that .02.
140 Prospective Teachers' Understanding of Division Then he laughed self-consciously, noting "the complex math terms that teach- ers use, like 'get rid of."' The other teacher candidates gave similar answers. They all talked about getting rid of the .2, isolating x, and multiplying both sides by .2. They seemed to see the question as quite straightforward and unproblematic, unlike some of the other questions they had been asked. Even when probed for reasons, they did not pro- vide conceptual justification for the procedure. I have no idea! Four elementary teacher candidates did not know how to solve the equation themselves. One was overwhelmed at the prospect of having to help a student solve an equation such as this one. "Oh, my God!" she exclaimed when I presented her with the question. She said she had no idea, although she knew "there's steps that you go through to do it." Another said she hadn't "done these" in so long that she just couldn't remember. All four of the teacher candidates who could not solve the equation attributed it to not having done algebra problems in a long time and not being able to remem- ber the procedures for solving equations such as this one. The only difference between these teacher candidates and the fourteen who focused on procedures was that these four could not remember the procedures. However, like the fourteen, they did not focus on meaning. DISCUSSION Although all but two prospective teachers could calculate 1/4 + V2 correctly, both the elementary and the mathematics majors had significant difficulty with the meaning of division with fractions. The difficulties experienced by all the teacher candidates (including those who succeeded in generating an appropriate represen- tation) indicated a narrow understanding of division. Most were able to consider division in partitive terms only: forming a certain number of equal parts. This model of division corresponds less easily to division with fractions than does the measurement interpretation of division. In a study of preservice elementary teach- ers' understanding of division, Graeber, Tirosh, and Glover (1986) also found that teacher candidates tended to think only in terms of this partitive interpretation. This finding offers some insight into why the task of making meaning out of 1/4+ V2 was so difficult for the prospective teachers in this study. Division by zero comes up frequently in college mathematics. The students who were majoring in mathematics had had more recent experience with dividing by zero than the other students. As such, it was not surprising that the secondary teacher candidates were better prepared to deal with this question, both in terms of providing mathematical explanations and in terms of knowing the correct rule, than were the elementary majors. Egg-Cartons of Mathematical Ideas: Fragmented Understandings The prospective teachers' understanding appeared to comprise remembering the rules for specific cases. Evidence for this conclusion is especially clear in the teacher candidates' efforts to generate representations for 13/4 -- /2. This task, un-
Deborah Loewenberg Ball 141 like the other two division questions, required them to do more than reproduce what they had been taught. Division with fractions is rarely taught conceptually in school; most of the prospective teachers probably learned to divide with fractions without necessarily thinking about what the problems meant. Indeed, most of them could carry out the procedure to produce the correct answer. Yet when they tried to generate a representation for the statement, most of them either represented 1/4 + 2 or could not do it at all. The results for this question suggest that, in most cases, the prospective teach- ers' understanding of division with fractions consisted of remembering a particu- lar rule; that understanding was unattached to other ideas about division. The re- sults for the other two questions are consistent with this interpretation. As with the division of fractions, most of the teacher candidates did not seem to refer to the more general concept of division or to multiplicative ideas and relationships to provide their explanations. Instead they recognized division by zero as a particu- lar case for which there was a rule. Their explanations were simply statements of what they though to be true for this specific case. Furthermore, half the elemen- tary candidates had the rule wrong. This analysis of the teacher candidates' understandings of division and of their ideas about what is entailed in explaining or justifying something mathematically fit with evidence from other parts of the interview that the prospective teachers' substantive understanding of mathematics tended to be both rule-bound and com- partmentalized (see Ball, 1988, in press). Although the three interview questions all dealt with division, the teacher candidates did not focus from case to case on the concept of division or on the multiplicative core of the questions. Instead, all but one student responded to each question in terms of the specific bit of mathe- matical knowledge entailed---division of fractions, division by zero, solving alge- braic equations involving division. What Does It Mean to Explain Something in Mathematics? For all three questions, the prospective teachers, both the mathematics majors and the elementary candidates, tended to search for the particular rules-"you can't divide by zero" or "get rid of the denominator"-rather than focusing on underlying meanings. They seemed to assume that stating a rule was tantamount to settling a mathematical question. In fact, within an intellectual community of mathematicians, providing a definition may provide one part of a mathematical ar- gument, for mathematicians share assumptions and basic definitions founded on convention or prior logical proof. These need not be unpacked and rejustified each time they arise. Classrooms, however, are emerging mathematical communities, in which shared assumptions and definitions are being constructed. If teachers are to be able to guide such constructions (see Lampert, in press), they must understand what is entailed in opening up and mathematically justifying those ideas and be able to do so. The prospective teachers' knowledge of division seemed founded more on memo- rization than on conceptual understanding. Some of the teacher candidates could
142 Prospective Teachers' Understanding of Division not remember the rules at all. Once forgotten, rules are not easily retrievable with- out the concepts to support them (Hiebert & Lefevre, 1986). Sheer memorization serves well to display mathematical knowledge in school-until one forgets, that is. The secondary teacher candidates, having had more (and more recent) opportu- nities to maintain their inventory of remembered knowledge, were more likely to have something to say and less likely to draw a complete blank. But few teacher candidates in either group were either disposed or able to provide mathematically legitimate explanations for the answers they gave. Was the students' apparent weakness in providing mathematically legitimate explanations influenced by the nature of the interview questions themselves? Two of the questions---division by zero and division in algebraic equations-were for- mulated in such a way that teacher candidates could simply retrieve the correct piece of information (e.g., "division by zero is undefined") as it was taught. These two questions examined conventionally packaged pieces of knowledge-knowl- edge that the teacher candidates had been taught in school. If they could remem- ber the necessary piece, they could answer each question by stating the rule. One might argue that nothing in either question or in the follow-up probes compelled them to talk about meaning or encouraged them to access legitimate explanations. Both questions, however, did ask the teacher candidates how they would respond to a pupil who raised that question, and probes pushed them to articulate underly- ing meanings and principles. Across the interviews, teacher candidates demon- strated that they wanted to give the pupils what they considered to be meaningful answers, but often they could not do so because their subject matter knowledge, lacking mathematical reason and meaning, was insufficient to act on that commit- ment (Ball, 1988). One of the mathematics majors realized this and commented (on division by zero), "I just know that...I don't really know why...it's almost become a fact...something that's just there." In answering the questions, many of the teacher candidates agonized over not having a concrete example or not knowing why something was true. One of the secondary candidates, for example, in answering the division by zero question, said she "would hate to say it is one of those things that you have to accept in math" but that she might have to in this case if she couldn't think of a concrete example. Another laughed wryly at himself for using the phrase "get rid of the denomina- tor," but did not have accessible any alternative ways of understanding. Making meaning out of division with fractions was a struggle for almost everyone. The rules the teacher candidates gave were what they remembered from what their teachers said. These results provided evidence that even correct answers were not undergirded with legitimate mathematical warrants or with understanding. The present study highlights that relying on what prospective teachers have learned in their precollege mathematics classes is unlikely to provide adequate subject matter preparation for teaching mathematics for understanding. Attending seriously to the subject matter preparation of elementary and secondary mathemat- ics teachers implies the need to know much more than we currently do about how teachers can be helped to transform and increase their understandings of mathe-
Deborah Loewenberg Ball 143 matics, working with what they bring and helping them move toward the kinds of mathematical understanding needed in order to teach mathematics well. REFERENCES Ball, D. L. (1988). Knowledge and reasoning in mathematical pedagogy: Examining what prospective teachers bring to teacher education. Unpublished doctoral dissertation. Michigan State University, East Lansing, MI. Ball, D. L. (in press). Research on teaching mathematics: Making subject matter knowledge part of the equation. In J. Brophy (Ed.), Advances in research on teaching, (Vol. 2). Greenwich, CT: JAI Press. Ball, D. L., & McDiarmid, G. W. (1988). Research on teacher learning: Studying how teachers' knowl- edge changes. Action in Teacher Education, 10(2), 14-21. Blaire, E. (1981). Philosophies of mathematics and perspectives of mathematics teaching. International Journal of Mathematics Education in Science and Technology, 12, 147-153. Davis, R. (1986). Conceptual and procedural knowledge in mathematics: summary analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 265-300). Hillsdale, NJ: Erlbaum. Ernest, P. (1988). The knowledge, beliefs, and attitudes of the mathematics teacher: A model. Unpub- lished manuscript, University of Exeter, England. Even, R. (1989). Prospective secondary teachers' knowledge of mathematicalfunctions. Unpublished dissertation, College of Education, Michigan State University, East Lansing, MI. Ferrini-Mundy, J. (1986, April). Mathematics teachers' attitudes and beliefs: Implications for in- service education. Paper presented at the annual meeting of the American Educational Research As- sociation, San Francisco. Graeber, A., Tirosh, D., & Glover, R. (1986, September). Preservice teachers' beliefs and perform- ance on measurement and partitive division problems. Paper presented at the eighth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Edu- cation, East Lansing, MI. Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge. In J. Hiebert (Ed.), Concep- tual and procedural knowledge: The case of mathematics (pp. 1-27). Hillsdale, NJ: Erlbaum. Kuhs, T. (1980). Elementary school teachers' conceptions of mathematics content as a potential influ- ence on classroom instruction. Unpublished doctoral dissertation, College of Education, Michigan State University, East Lansing, MI. Lampert, M. (in press). Reinventing the meaning of mathematical knowing in the classroom. Ameri- can Educational Research Journal. Leinhardt, G., & Smith, D. (1985). Expertise in mathematics instruction: Subject matter knowledge. Journal of Educational Psychology, 77, 247-271. Lerman, S. (1983). Problem-solving or knowledge-centred: The influence of philosophy on mathemat- ics teaching. International Journal of Mathematics, Education, Science, and Technology, 14(1), 59-66. National Center for Research on Teacher Education. (1988). Teacher education and learning to teach: A research agenda. Journal of Teacher Education, 32(6), 27-32. Owens, J. E. (1987). A study offour preservice secondary mathematics teachers' constructs of mathe- matics and mathematics teaching. Unpublished doctoral dissertation, University of Georgia, Athens, GA. Peterson, P., Fennema, E., Carpenter, T., & Loef, M. (1989). Teachers' pedagogical content beliefs in mathematics. Cognition and Instruction 6(1), 1-40. Post, T., Harel, G., Behr, M. & Lesh, R. (1988). Intermediate teachers' knowledge of rational number concepts. In E. Fennema, T. Carpenter & S. Lamon (Eds.), Integrating research on teaching and learning mathematics (pp. 194-219). Madison, WI: Wisconsin Center for Education Research., University of Wisconsin. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Re- searcher, 15(2), 4-14.
144 Prospective Teachers' Understanding of Division Steinberg, R., Haymore, J., & Marks, R. (1985, April). Teachers' knowledge and content structuring in mathematics. Paper presented at the annual meeting of the American Educational Research As- sociation, Chicago. Thompson, A. (1984). The relationship of teachers' conceptions of mathematics and mathematics teaching to instructional practice. Educational Studies in Mathematics, 15, 105-127. AUTHOR DEBORAH LOEWENBERG BALL, Assistant Professor, Department of Teacher Education, Michi- gan State University, 116 Erickson Hall, East Lansing, MI48824.
Journal for Research in Mathematics Education 1990, Vol. 21, No. 2, 132-144 PROSPECTIVE ELEMENTARY AND SECONDARY TEACHERS' UNDERSTANDING OF DIVISION DEBORAH LOEWENBERG BALL, Michigan State University This article reports an analysis of 19 prospective elementary and secondary teachers' understanding of division. Interview questions probed the prospective teachers' understanding of division in three contexts. Although many of the teacher candidates could produce correct answers, several could not, and few were able to give mathematical explanations for the underlying principles and meanings. The prospective teachers' knowledge was generally fragmented, and each case of division was held as a separate bit of knowledge. Teachers' subject matter knowledge is enjoying a renaissance in research on teaching (Shulman, 1986) and teacher education (National Center for Research on Teacher Education, 1988). From a range of perspectives and with a variety of approaches, researchers are focusing increasingly on the subject matter knowledge of teachers and its role in teaching. In research on the teaching of mathematics, some researchers investigate teachers' and prospective teachers' beliefs about the subject or their ideas about teaching it (Blaire, 1981; Ernest, 1988; Ferrini-Mundy, 1986; Kuhs, 1980; Lerman, 1983; Peterson, Fennema, Carpenter, & Loef, 1989; Thompson, 1984). These studies generally highlight the influence of teachers' assumptions about mathematics on their teaching of the subject. Other researchers focus on teachers' and prospective teachers' understandings of specific topics (Ball, 1988; Ball & McDiarmid, 1988; Davis, 1986; Even, 1989; Leinhardt & Smith, 1985; Owens, 1987; Post, Behr, Harel, & Lesh, 1988; Steinberg, Haymore, & Marks, 1985). They explore how teachers think about their mathematical knowl- edge and how they understand (or misunderstand) specific ideas. The analysis reported in this article is part of a larger study that examined pro- spective teachers' knowledge and reasoning, both mathematical and pedagogical, at the point when they entered teacher education (Ball, 1988). This article focuses on one aspect of the teacher candidates' substantive knowledge of mathematics, their understanding of division. The article also examines what they believed makes something true or reasonable in mathematics, what counts as a mathemati- cal justification. This research was supported in part by the National Center for Research on Teacher Edu- cation, Michigan State University. The NCRTE is funded primarily by the Office of Educa- tional Research and Improvement, United States Department of Education. The opinions expressed herein are those of the author and do not necessarily reflect the position, policy, or endorsement of the OERI/ED. (Grant No. OERI-G-86-0001) The author gratefully acknowledges Ruhama Even, Mary M. Kennedy, Magdalene Lam- pert, Penelope Peterson, and Suzanne M. Wilson, as well as the JRME editor and reviewers, for their helpful comments on this article.
133 METHOD Participants Ten elementary and nine secondary education students were interviewed when they were about to enroll for their first education course. The prospective elemen- tary teachers were majoring in elementary education or child development and had no disciplinary specialization. The secondary teacher education students were mathematics majors or minors. The sample of prospective teachers was systematically selected to vary on sev- eral key criteria: gender, academic history in college mathematics, and self-ac- knowledged dispositions toward mathematics. Of the 19 students, 6 were men and 13 were women. One student was black, one was Asian; the others were Cauca- sian. Although the group was not selected to be representative, the demographic data indicated that the sample was similar to the population of teacher education students in terms of characteristics such as ethnicity, gender, social class, and age. Student academic records were an additional source of data about the prospec- tive teachers selected to participate in the study. Analyses of academic records revealed three high school valedictorians within the sample (one elementary and two secondary majors). On a 4-point scale, the average overall grade-point aver- age of the elementary majors was 3.18; the secondary majors averaged 3.09. The mean grade-point average for mathematics courses was 3.12 and 3.13, respec- tively. The elementary teacher candidates had taken, on average, fewer than two mathematics courses since high school, and these were often remedial. The secon- dary teacher candidates had taken an average of 32 quarter credits, or more than nine college-level mathematics courses. Instrumentation Three different mathematical contexts were employed to examine prospective teachers' knowledge of division: division with fractions, division by zero, and division with algebraic equations. In each case, the teacher candidates were asked to explain or to generate representations. Probes were developed to examine the teacher candidates' ideas about what it means to justify or to explain something in mathematics. The three questions were posed in the order reported in this article and were not separated by other questions or tasks. The interview questions and standard probes are reported below. Additional probes were generated in the course of the individual interviews when it was nec- essary in order to get clearer about what the person was thinking. In all cases, ef- fort was directed at probing as deeply as possible the prospective teacher's under- standing of the mathematical ideas and of what counts as a mathematically reason- able justification. Procedures The interviews were tape-recorded and transcribed. Careful substantive analy- ses of the interview questions led to the creation of a set of response categories for
134 Prospective Teachers' Understanding of Division each one. These categories were modified in the course of data analysis to better accommodate teacher candidates' responses. In the larger study, most questions were cross-analyzed on several dimensions: subject matter understanding; ideas about teaching; learning and the teacher's role; and feelings or attitudes about mathematics, pupils, or self. For the present analysis of prospective teachers' knowledge of division, responses were coded on two dimensions: (1) their correct- ness and (2) the nature of the justification provided by the teacher candidate. RESULTS Division By Fractions Participants were asked the following: People have different approaches to solving problems involving division with frac- tions. How would you solve this one: Sometimes teachers try to come up with real-world situations or story problems to show the meaning or application of some particular piece of content. This can be pretty challenging to do. What would you say would be a good situation or story for 1/4 + 2-something real for which 1/4 + /2 is the appropriate mathematical formulation? After the participant described a situation or story, he or she was asked: How does that fit with 13A + /A? If the participant noticed that the answer to the story or other representation did not match the answer obtained computationally, he or she was asked: Why did that come out different? If the participant struggled with finding a representation, or could not do it, he or she was asked: Many people find this hard. In your view, what makes this difficult? The teacher candidates' responses to being asked to solve and to develop a rep- resentation for 13/4 - 1/2 suggested that they perceived the task to be about fractions, not division. When asked, for example, what made this difficult, most commented that it was hard (or impossible) to relate 1/4 + 1/2 to real life because, as one said, "you don't think in fractions, you think more in whole numbers." Not only did their explanations reveal that they framed the problem in terms of fractions, but also that many were uncomfortable with fractions as quantities. Several com- mented that they did not like fractions. Seventeen out of 19 participants were able to calculate the division correctly, but only 5 were able to generate an appropriate representation. Their responses are summarized in Table 1. Appropriate representations. Five teacher candidates, all mathematics majors, generated an appropriate representation of 1/4 + V2; however, this did not come easily to any of them. For example, Terrell first said he couldn't "think of anything
Deborah Loewenberg Ball 135 Table 1 Prospective Elementary and Secondary Teachers' Calculations and Representations of 1 3/4 + 1/2 Elementary Secondary Totals Correct calculation 9 8 17 Appropriate representation 0 5 5 Inappropriate representation 3 2 5 Unable to generate a representation 6 2 8 Note. There were 10 elementary and 9 secondary teacher candidates. One elementary candidate was not asked this question. specific." Then he said he would use pizza: If you took the pizza and took one-half of a pizza and you took a whole pizza and three- quarters of a pizza, that would be 11/4. You put the one-half of the pizza on top of each piece. So first you'd take the whole pizza and you'd put it on top of it. Then you'd take that off, whatever it fits on and you'd do it again. Only take it off if it fits the whole thing. If...both pieces are equal. Then you go through the one-half a piece and do the same for that. Take that off. Then you get that last piece and you...well, that's the way I'd explain it. Terrell then explained what the answer (31/2) meant in this context: You'd take the one-half and the answer would be how many times you got a whole half (if you want to say that). Of the...whatever's left over, what part of it is of the half, I guess you could say. You'd have a V4 left, which is one-half of a half. While sometimes rather confusing to follow, these prospective teachers' re- sponses did make mathematical sense. Their answers indicated that, using the measurement model of division, they were thinking in terms of how many halves there are in one and three-fourths. Inappropriate representations. Five teacher candidates generated representa- tions that did not correspond to 1 /4+ ?. The most frequent error was to represent division by two instead of division by one-half. For example, Barb, a mathematics major, gave the following story: If we had one and three-quarters pizzas left and there were two of us dying to split it, then how would we be able to split that? Answering her own questions, she said each person would get 7/8 of a pizza alto- gether. This error, representing division by two rather than division by one-half, was the most common error made by the participants. Although the answer they got to their story problem (7/8) differed from their cal- culated answer (3?), the four prospective teachers did not seem to notice that they were dividing in half instead of by one-half. The discrepancy seemed to be masked for them by a change in the referent unit from whole pizzas to fourths of pizzas. Here is a typical example of the students' reasoning: "I have 13/ pizzas which I want to split equally between two hungry teenagers (1/4 + 2). Each pizza is divided into four pieces, so you have seven pieces. So each person gets 7/ of a pizza, which is 3? pieces of pizza." Allen, an elementary major with 27 credits in college mathematics (through the
136 Prospective Teachers' Understanding of Division calculus course sequence) seemed to get stuck by his knowlege of the algorithm "invert and multiply": Somebody has one and three quarters apples or something like that and they wanted to double up their, double the amount of apples they have, um, just give 'em an equation...using only fractions. Other than that I couldn't think of any, any situation where you could, where you would logically divide a number like that by one half instead of just multiplying by two. You would have to have, be, be working on, on dividing like fractions and setting up equations using that. Right, you just have to say...."Well, you know, use this and figure the story problem out but, only use frac- tions." Allen's story modeled 1/4 x 2-the common procedure used in fraction division. Using "invert and multiply" as his frame of reference, Allen did not seem to focus on the concept of division by one-half. No representation. Eight teacher candidates could not generate any representa- tion, correct or incorrect, for 13/4 + 1/2. Marsha, for example, said she felt stuck and couldn't even remember how to get the answer. She explained that she hadn't done this since high school: Egad, I don't even remember how to do this. (pause) Shoot, no, I have to, I don't know how to do it. Because I don't know what I did ...I found the common denominator and I think what I have to do is go four times one is four and then plus three is seven- fourths, no I think that's what I did, one-half, but then, see, I don't know what I need to divide. I don't even remember that...I remember doing these for a long time though and trying to get these down, and so I remember bits and pieces and then I try to apply it generally, and I can't do it. The prospective teachers who did not generate a representation at all seemed to fall into two groups. Two participants (one elementary, one secondary) did recog- nize the conceptual problem. They initially proposed stories or models that repre- sented division by two and then realized that they were representing division in half, not by one-half. The other six participants, however, seemed to think that trying to relate 1/4 + V2 to a concrete situation was not a feasible task-that 1/4 + /2 could not be represented in real-world terms. The two participants who recognized that division by one-half is not the same as division by two revealed a better grasp of the idea than those who constructed a story that represented division by two. Still, despite this recognition, they were unable to figure out what division by one-half meant. Those who thought it was an impossible task may have been revealing a view of mathematics as a senseless activity, out of which meaning cannot necessarily be made. Division by Zero Participants were asked the following questions: Suppose that a student asks you what 7 divided by 0 is. How would you respond? Why is that what you'd want to say? Following whatever the participant said, he or she was probed: Suppose the student asks you why that is? If a participant said, "I'd say it's undefined," he or she was asked:
Deborah Loewenberg Ball 137 What if the student asks, "What do you mean by 'undefined'?" If a participant said, "I'd say you can't divide by 0," he or she was asked: What if a student asks, "Why can't you divide by zero?" If a participant said they would show students how, as you divide by smaller and smaller numbers, the answer gets larger and larger, he or she was asked: What would I see or hear you doing? Of the 19 teacher candidates, 5 explained the meaning of division by zero in response to this question. Twelve of the prospective teachers responded by stating rules, five of which were incorrect. Two participants did not know. (See Table 2 for the distribution of types of responses.) Each category is discussed below. Table 2 Prospective Elementary and Secondary Teachers' Explanations for Division by Zero Elementary Secondary Totals Meaning 1 4 5 Correct rule 2 5 7 Incorrect rule 5 0 5 Don't know 2 0 2 Note. There were 10 elementary and 9 secondary teacher candidates. Explanations focused on meaning. Four teacher candidates gave answers that fo- cused on what division by zero means. Two approaches were used: (1) showing that division by zero was undefined and (2) showing that the quotient "explodes" as the divisor decreases. Tim, a mathematics major, chose the first approach. He said that he would write 7 + 0 "in mathematical form" on the board-i.e., with a division bracket: 0)7. Then he would explain that: You cannot divide seven by zero because there is nothing multiplied by zero to get seven. In other words, everything multiplied by a zero is zero, so if we had seven over zero. Okay, if we had zero divided by seven there is nothing multiplied, there is no number up here you could put to get zero. There's no number you can put up here to get seven. And I would show them that. Whereas six divided by two there's a number you can put up there. And whenever you come across that case, you can't find a num- ber to put up there, it doesn't exist, you can't do it. Allen, an elementary major, explained division by zero using the second ap- proach: Dividing seven by three and then divide seven by two and then divide seven by one and, uh, and when they get up to seven divided by one is seven and you were to go one step farther you'd, you'd have numbers that were keep getting larger and larger....Um, one step farther you would have to, you know, say divide it by zero....I guess it would be better to start getting closer to zero using the decimals and see that dividing seven by fractions makes numbers, you know, they keep getting larger and larger and, uh, that if you keep making that, the divisor closer and closer to zero, the number's just gonna keep getting larger and larger and larger and, uh, then I'd start asking them what the largest number they can think of is so then, that there is no largest number that, uh, that it, there is really no such statement as seven divided by zero. Mark and Allen both focused specifically on the case of dividing by zero. What
138 Prospective Teachers' Understanding of Division their answers had in common was the aim of showing why the particular case of division by zero is impossible. You can't divide by zeto. Seven teacher candidates explained division by zero in terms of a rule such as "you can't divide by zero." Unlike those who focused on meaning, these prospective teachers did not try to show why this was so. Instead they emphasized the importance of remembering the rule. Terrell, a mathematics major, said emphatically, I'd just say..."It's undefined," and I'd tell them that this is a rule that you should never forget that anytime you divide by zero, you can't. You just can't. It's undefined, so...you just can't. He added, "Anytime you get a number divided by zero, then you did something wrong before." Andy, another mathematics major, said, "You can't divide by zero...It's just something to remember." Cindy, also majoring in mathematics, said she would tell students that "this is something that you won't ever be able to do in mathematics"--even in calculus. Even when probed, these students did not provide a mathematical justification for the principle that division by zero was not permitted. Anything divided by zero is zero. Five other teacher candidates also responded in term of a rule. Like the prospective teachers quoted above, their notions of mathematical explanation seemed to mean restating rules. What made their re- sponses different, however, was that the rule they invoked was not true. Linda, an elementary major, was perhaps the most emphatic: I'd just say, "Anything divided by zero is zero. That's just a rule, you just know it." Or I'd say, "Well, if you don't have anything, you can't get anything out. You know, it's empty, it's nothing, so you can't get anything out of it." Anything multiplied by zero is zero. I'd just say, "That's something that you have to learn, you have to know." I think that's how I was told. You just know it...I'd just say, you know, if they were older and they asked me "Why?" I'd just have to start mumbling about something, I don't know...I don't know what. I'd just tell them "Because!" (laughs) That's just the way it is...that's just one of those rules...something like that...you know, in English...sometimes the C sounds like K or...you know, you just learn. I before E ex- cept after C, one of those things, in my view. Interestingly, although Linda mentioned multiplication by zero, she didn't con- nect it with the problem of dividing by zero. Like those who stated "you can't divide by zero," these prospective teachers all emphasized the absoluteness of the rule and the value of getting pupils to remem- ber it. Explaining and knowing mathematics was centered on rules. However, these teacher candidates did not realize that their rules were invalid. I don't remember. Two prospective elementary teachers said they could not remember the answer to 7 + 0. Mei Ling said simply, "Seven divided by zero? Isn't that-isn't there a term for the answer to that? I can't remember." Rachel, who had taken a little more mathematics, more recently and more successfully than most of the other elementary majors, was simply stumped by this question. "Seven divided by zero," she mused. "I'm having trouble...is that zero or is that seven? I'm trying to think myself." Because these students could not remember the answer
Deborah Loewenberg Ball 139 to 7 + 0, their ideas about what would count as a reasonable explanation were not revealed. Algebraic Equations Participants were asked: Suppose that one of your students asks you for help with the following: x If = 5, then x = 0.2 How would you respond? Why is that what you'd do? If a participant said, "I'd tell them to do the same thing to both sides," or "I'd tell them to multiply both sides by 0.2," he or she was asked: What if the student asks, "Why do you do that?" In response to this question involving division in algebra, the teacher candidates focused on the mechanics of solving algebraic equations. Overwhelmingly they "explained" x -=5 0.2 by restating the steps of procedures to solve such equations. Only one prospective teacher talked about it in terms of what it meant, and a few teacher candidates didn't know how to do it at all. (See Table 3.) Table 3 Prospective Elementary and Secondary Teachers' Explanations for Division in Algebra Elementary Secondary Totals Meaning 1 0 1 Procedure 5 9 14 Don't know 4 0 4 Note. There were 10 elementary and 9 secondary teacher candidates. Focus on meaning. Only one teacher candidate-an elementary major-tried to talk about the meaning of the equation. Sandi said that she would want the pupil "to understand what he's doing first." She said she would help the pupil under- stand "the idea that the .2 has to go into x." Although her explanation was vague, she was trying to make sense of the problem by reasoning about division. Focus on mechanics. Fourteen of the prospective teachers, including all of the mathematics majors, focused on the mechanics of manipulating algebraic equa- tions. Terrell, a secondary candidate, said I'd explain that somehow you have to get this x by itself without that .02, I mean 0.2...and then I'd ask her, I'd ask her...I'd tell her somehow she's going to have to get rid of that .02.
140 Prospective Teachers' Understanding of Division Then he laughed self-consciously, noting "the complex math terms that teach- ers use, like 'get rid of."' The other teacher candidates gave similar answers. They all talked about getting rid of the .2, isolating x, and multiplying both sides by .2. They seemed to see the question as quite straightforward and unproblematic, unlike some of the other questions they had been asked. Even when probed for reasons, they did not pro- vide conceptual justification for the procedure. I have no idea! Four elementary teacher candidates did not know how to solve the equation themselves. One was overwhelmed at the prospect of having to help a student solve an equation such as this one. "Oh, my God!" she exclaimed when I presented her with the question. She said she had no idea, although she knew "there's steps that you go through to do it." Another said she hadn't "done these" in so long that she just couldn't remember. All four of the teacher candidates who could not solve the equation attributed it to not having done algebra problems in a long time and not being able to remem- ber the procedures for solving equations such as this one. The only difference between these teacher candidates and the fourteen who focused on procedures was that these four could not remember the procedures. However, like the fourteen, they did not focus on meaning. DISCUSSION Although all but two prospective teachers could calculate 1/4 + V2 correctly, both the elementary and the mathematics majors had significant difficulty with the meaning of division with fractions. The difficulties experienced by all the teacher candidates (including those who succeeded in generating an appropriate represen- tation) indicated a narrow understanding of division. Most were able to consider division in partitive terms only: forming a certain number of equal parts. This model of division corresponds less easily to division with fractions than does the measurement interpretation of division. In a study of preservice elementary teach- ers' understanding of division, Graeber, Tirosh, and Glover (1986) also found that teacher candidates tended to think only in terms of this partitive interpretation. This finding offers some insight into why the task of making meaning out of 1/4+ V2 was so difficult for the prospective teachers in this study. Division by zero comes up frequently in college mathematics. The students who were majoring in mathematics had had more recent experience with dividing by zero than the other students. As such, it was not surprising that the secondary teacher candidates were better prepared to deal with this question, both in terms of providing mathematical explanations and in terms of knowing the correct rule, than were the elementary majors. Egg-Cartons of Mathematical Ideas: Fragmented Understandings The prospective teachers' understanding appeared to comprise remembering the rules for specific cases. Evidence for this conclusion is especially clear in the teacher candidates' efforts to generate representations for 13/4 -- /2. This task, un-
Deborah Loewenberg Ball 141 like the other two division questions, required them to do more than reproduce what they had been taught. Division with fractions is rarely taught conceptually in school; most of the prospective teachers probably learned to divide with fractions without necessarily thinking about what the problems meant. Indeed, most of them could carry out the procedure to produce the correct answer. Yet when they tried to generate a representation for the statement, most of them either represented 1/4 + 2 or could not do it at all. The results for this question suggest that, in most cases, the prospective teach- ers' understanding of division with fractions consisted of remembering a particu- lar rule; that understanding was unattached to other ideas about division. The re- sults for the other two questions are consistent with this interpretation. As with the division of fractions, most of the teacher candidates did not seem to refer to the more general concept of division or to multiplicative ideas and relationships to provide their explanations. Instead they recognized division by zero as a particu- lar case for which there was a rule. Their explanations were simply statements of what they though to be true for this specific case. Furthermore, half the elemen- tary candidates had the rule wrong. This analysis of the teacher candidates' understandings of division and of their ideas about what is entailed in explaining or justifying something mathematically fit with evidence from other parts of the interview that the prospective teachers' substantive understanding of mathematics tended to be both rule-bound and com- partmentalized (see Ball, 1988, in press). Although the three interview questions all dealt with division, the teacher candidates did not focus from case to case on the concept of division or on the multiplicative core of the questions. Instead, all but one student responded to each question in terms of the specific bit of mathe- matical knowledge entailed---division of fractions, division by zero, solving alge- braic equations involving division. What Does It Mean to Explain Something in Mathematics? For all three questions, the prospective teachers, both the mathematics majors and the elementary candidates, tended to search for the particular rules-"you can't divide by zero" or "get rid of the denominator"-rather than focusing on underlying meanings. They seemed to assume that stating a rule was tantamount to settling a mathematical question. In fact, within an intellectual community of mathematicians, providing a definition may provide one part of a mathematical ar- gument, for mathematicians share assumptions and basic definitions founded on convention or prior logical proof. These need not be unpacked and rejustified each time they arise. Classrooms, however, are emerging mathematical communities, in which shared assumptions and definitions are being constructed. If teachers are to be able to guide such constructions (see Lampert, in press), they must understand what is entailed in opening up and mathematically justifying those ideas and be able to do so. The prospective teachers' knowledge of division seemed founded more on memo- rization than on conceptual understanding. Some of the teacher candidates could
142 Prospective Teachers' Understanding of Division not remember the rules at all. Once forgotten, rules are not easily retrievable with- out the concepts to support them (Hiebert & Lefevre, 1986). Sheer memorization serves well to display mathematical knowledge in school-until one forgets, that is. The secondary teacher candidates, having had more (and more recent) opportu- nities to maintain their inventory of remembered knowledge, were more likely to have something to say and less likely to draw a complete blank. But few teacher candidates in either group were either disposed or able to provide mathematically legitimate explanations for the answers they gave. Was the students' apparent weakness in providing mathematically legitimate explanations influenced by the nature of the interview questions themselves? Two of the questions---division by zero and division in algebraic equations-were for- mulated in such a way that teacher candidates could simply retrieve the correct piece of information (e.g., "division by zero is undefined") as it was taught. These two questions examined conventionally packaged pieces of knowledge-knowl- edge that the teacher candidates had been taught in school. If they could remem- ber the necessary piece, they could answer each question by stating the rule. One might argue that nothing in either question or in the follow-up probes compelled them to talk about meaning or encouraged them to access legitimate explanations. Both questions, however, did ask the teacher candidates how they would respond to a pupil who raised that question, and probes pushed them to articulate underly- ing meanings and principles. Across the interviews, teacher candidates demon- strated that they wanted to give the pupils what they considered to be meaningful answers, but often they could not do so because their subject matter knowledge, lacking mathematical reason and meaning, was insufficient to act on that commit- ment (Ball, 1988). One of the mathematics majors realized this and commented (on division by zero), "I just know that...I don't really know why...it's almost become a fact...something that's just there." In answering the questions, many of the teacher candidates agonized over not having a concrete example or not knowing why something was true. One of the secondary candidates, for example, in answering the division by zero question, said she "would hate to say it is one of those things that you have to accept in math" but that she might have to in this case if she couldn't think of a concrete example. Another laughed wryly at himself for using the phrase "get rid of the denomina- tor," but did not have accessible any alternative ways of understanding. Making meaning out of division with fractions was a struggle for almost everyone. The rules the teacher candidates gave were what they remembered from what their teachers said. These results provided evidence that even correct answers were not undergirded with legitimate mathematical warrants or with understanding. The present study highlights that relying on what prospective teachers have learned in their precollege mathematics classes is unlikely to provide adequate subject matter preparation for teaching mathematics for understanding. Attending seriously to the subject matter preparation of elementary and secondary mathemat- ics teachers implies the need to know much more than we currently do about how teachers can be helped to transform and increase their understandings of mathe-
Deborah Loewenberg Ball 143 matics, working with what they bring and helping them move toward the kinds of mathematical understanding needed in order to teach mathematics well. REFERENCES Ball, D. L. (1988). Knowledge and reasoning in mathematical pedagogy: Examining what prospective teachers bring to teacher education. Unpublished doctoral dissertation. Michigan State University, East Lansing, MI. Ball, D. L. (in press). Research on teaching mathematics: Making subject matter knowledge part of the equation. In J. Brophy (Ed.), Advances in research on teaching, (Vol. 2). Greenwich, CT: JAI Press. Ball, D. L., & McDiarmid, G. W. (1988). Research on teacher learning: Studying how teachers' knowl- edge changes. Action in Teacher Education, 10(2), 14-21. Blaire, E. (1981). Philosophies of mathematics and perspectives of mathematics teaching. International Journal of Mathematics Education in Science and Technology, 12, 147-153. Davis, R. (1986). Conceptual and procedural knowledge in mathematics: summary analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 265-300). Hillsdale, NJ: Erlbaum. Ernest, P. (1988). The knowledge, beliefs, and attitudes of the mathematics teacher: A model. Unpub- lished manuscript, University of Exeter, England. Even, R. (1989). Prospective secondary teachers' knowledge of mathematicalfunctions. Unpublished dissertation, College of Education, Michigan State University, East Lansing, MI. Ferrini-Mundy, J. (1986, April). Mathematics teachers' attitudes and beliefs: Implications for in- service education. Paper presented at the annual meeting of the American Educational Research As- sociation, San Francisco. Graeber, A., Tirosh, D., & Glover, R. (1986, September). Preservice teachers' beliefs and perform- ance on measurement and partitive division problems. Paper presented at the eighth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Edu- cation, East Lansing, MI. Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge. In J. Hiebert (Ed.), Concep- tual and procedural knowledge: The case of mathematics (pp. 1-27). Hillsdale, NJ: Erlbaum. Kuhs, T. (1980). Elementary school teachers' conceptions of mathematics content as a potential influ- ence on classroom instruction. Unpublished doctoral dissertation, College of Education, Michigan State University, East Lansing, MI. Lampert, M. (in press). Reinventing the meaning of mathematical knowing in the classroom. Ameri- can Educational Research Journal. Leinhardt, G., & Smith, D. (1985). Expertise in mathematics instruction: Subject matter knowledge. Journal of Educational Psychology, 77, 247-271. Lerman, S. (1983). Problem-solving or knowledge-centred: The influence of philosophy on mathemat- ics teaching. International Journal of Mathematics, Education, Science, and Technology, 14(1), 59-66. National Center for Research on Teacher Education. (1988). Teacher education and learning to teach: A research agenda. Journal of Teacher Education, 32(6), 27-32. Owens, J. E. (1987). A study offour preservice secondary mathematics teachers' constructs of mathe- matics and mathematics teaching. Unpublished doctoral dissertation, University of Georgia, Athens, GA. Peterson, P., Fennema, E., Carpenter, T., & Loef, M. (1989). Teachers' pedagogical content beliefs in mathematics. Cognition and Instruction 6(1), 1-40. Post, T., Harel, G., Behr, M. & Lesh, R. (1988). Intermediate teachers' knowledge of rational number concepts. In E. Fennema, T. Carpenter & S. Lamon (Eds.), Integrating research on teaching and learning mathematics (pp. 194-219). Madison, WI: Wisconsin Center for Education Research., University of Wisconsin. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Re- searcher, 15(2), 4-14.
144 Prospective Teachers' Understanding of Division Steinberg, R., Haymore, J., & Marks, R. (1985, April). Teachers' knowledge and content structuring in mathematics. Paper presented at the annual meeting of the American Educational Research As- sociation, Chicago. Thompson, A. (1984). The relationship of teachers' conceptions of mathematics and mathematics teaching to instructional practice. Educational Studies in Mathematics, 15, 105-127. AUTHOR DEBORAH LOEWENBERG BALL, Assistant Professor, Department of Teacher Education, Michi- gan State University, 116 Erickson Hall, East Lansing, MI48824.